We have seen that what may appear at first sight to be Inference to the Best Explanation can turn out to be describable with nothing more than the standard theories of probabilistic reasoning. It is no surprise, then, that some defenders of IBE have tried to set up an alliance between IBE and probabilistic reasoning. The form this takes in Lipton (2004 [72], pp. 103- 120) is a set of arguments that set out to establish that IBE is a useful addition to Bayesian reasoning. Day and Kincaid (1994 [20], p. 286) also state this possibility: “IBE can be embedded in determining the priors and likelihoods that make up Bayesian calculation”. Lipton goes into more detail, so it is his version I will discuss.13
At first sight, Bayesianism leaves no room for IBE. We start out with a probability distribution over all propositions. We get some new evidence and, using conditional probabilities, we update the probability distribution. This is our new state of belief. Explanatory considerations do not play a role.
But let us not be too hasty. Lipton proposes three ways in which explana- tory considerations might be hidden at the heart of the Bayesian machinery (2004 [72], p. 114). First, explanatory considerations might figure in the de- termination of likelihoods. Second, explanatory considerations might figure in the determination of priors. Third, explanatory considerations might fig- ure in the determination of relevant evidence. We need to look at these three suggestions in turn.
The first suggestion is that explanatory loveliness is correlated withlike- lihood. Likelihood is a technical term that designates the conditional prob- ability of the evidence on the hypothesis. Thus, if the hypothesis H is that the diners ate poisonous mushrooms, and the evidence E is that they both fell ill, then the likelihood of the evidence isP(E|H), which is the probabil- ity of falling ill when you have eaten poisonous mushrooms. Lipton (2004 [72], p. 114) suggests that “although likelihood is not to be equated with loveliness, it might yet be that one way we judge how likely E is on H is by considering how well H would explain E.” This would be pragmatically useful “if in fact loveliness is reasonably well correlated with likelihood, and we find it easier in practice to judge loveliness than likelihood.”
13See Hitchcock 2007 [48] and Psillos 2007 [95] for other appraisals of Lipton’s attempt
to reconcile IBE and Bayesianism. Hitchcock argues that, although such a reconciliation is probably possible, it cannot go through the explanatory virtue of simplicity. (However, I argue above that simplicity is not an explanatory virtue.) Psillos argues that defenders of IBE should be the enemies rather than the friends of Bayesianism, and suggests that Lipton yields too much ground to the Bayesians. See Lipton 2007 [73] for replies.
This proposal might be a good argument for IBE, if it were to be found correct. In order to justify IBE, we must show thatP(H|E) (the probability of H on E) is positively correlated with how lovely an explanation of E is given by H. Now, suppose we can show – as Lipton hopes – that loveliness is positively correlated with the likelihoodP(E|H). Then, using the rules of probability and the two further assumptions that loveliness is not negatively correlated with P(H) or positively correlated with P(E) (which would have to be independently argued for), we can show that loveliness is positively correlated with P(H|E). This is exactly what we need to prove in order to vindicate IBE.
But do we base our estimates of likelihood on the explanatory loveliness of H for E? What is it that we do when we desire to know the probabil- ity that, say, people fall ill after eating poisonous mushrooms? We attempt to gather the relevant empirical evidence. Experience, not explanatory con- siderations, teaches us whether in 95% or 30% or 4% of the cases eating poisonous mushrooms leads to illness.14
Let me elucidate this by an example where it is clear that (1) we know nothing at all about likelihoods, yet (2) we can easily judge loveliness. The following explanation of the illness of our mushroom gatherers is put forth: all the mushroom they gathered were edible, but as they walked through the woods, they were made the unwitting test subjects of an experimental non-lethal weapon mounted on a CIA satellite. This weapon sends out a strongly focused gravitational wave that, with probability p, causes sudden movements in the digestive system that allow more gastric acid to enter the duodenum than is normal. After a couple of hours, this leads to pains and illness.
This explanation is highly unlikely. It is also quite lovely: if it were true, it would be a good explanation of the illness of the mushroom gatherers. But is there anything, anything at all except that it is non-zero, that we can say about p once we have seen that this explanation is quite lovely? Does the loveliness of the explanation allow us to draw any conclusions about the success rate of a fictional CIA weapon? It is unlikely.
Thus, our examples cast doubt on the idea that there is a useful link between loveliness and likelihood. This does not prove that there is no such link; but since we have not been given any concrete arguments to believe that there is, it casts the ball back to the defenders of IBE.
One final note. There are some cases in which likelihood and lovelinessare
14Unless we define ‘poisonous’ in terms of the percentage of cases that lead to illness,
in which case it is not experience, but linguistic analysis or stipulation that gives us the correct percentage.
correlated: namely, those cases where we add detail to an explanation with the express purpose of pointing out the scenario with the greatest likelihood. For instance, we can simultaneously increase the loveliness and the likelihood of the hypothesis in the mushroom example by adding that the mushrooms weren’t just poisonous, they were in fact devil’s boletes, and that devil’s bo- letes contain a poison called bolesatine which causes gastrointestinal distress. But such cases are no comfort to the defenders of IBE. The probability of the (more detailed) devil’s boletes hypothesis is necessarily less than the proba- bility of the (less detailed) poisonous mushroom hypothesis. Thus, in these cases loveliness, although positively correlated with likelihood, is negatively correlated with probability – which defeats the purpose of the proposal.
A second place where explanatory considerations might play a role is in the determination of priors. Lipton proposes one mechanism by which this could take place: the prior probabilities of hypotheses might be decided on the basis of explanatory virtues like simplicity. That brings us right back to the argument from explanatory virtues, which we have already discussed in section 3.5.
The third possibility is that explanatory considerations are used todeter- mine the relevant evidence. Lipton (2004 [72], p. 116) writes: “Bayes’s theorem . . . does not, however, saywhich evidence one ought to conditionalise on. In principle, perhaps, non-demonstrative inference should be based on ‘total evidence’, indeed on everything that is believed. In practice, however, investigators must think about which bits of what they know really bear on their question, and they also need to decide which further observations would be particularly relevant. . . . [T]his seems yet another area where the expla- nationist may contribute. . . . [W]e sometimes come to see that a datum is epistemically relevant to a hypothesis precisely by seeing that the hypothesis would explain it.”
Lipton is onto something here – thinking about explanation does help us when we decide which evidence to take into account and what further exper- iments to perform. But notice two things. First, we are not talking about explanatory loveliness, but about whether a hypothesis explains something or not. This is an absolute rather than a comparative judgment. Second, on this proposal explanatory considerations do not influence our judgments of likeliness directly, but only by telling us what evidence we ought to gather and consider. This role for explanation in scientific method is not the role postulated by IBE; so I can (and do) grant Lipton’s claim here without granting that it constitutes a defence of IBE.
What we have seen, then, is that the attempt to tack IBE onto Bayesian- ism fails; or at least, that we do not at this moment have any reason to believe that it succeeds. Since this was the final class of arguments to be considered,
we can conclude that the current defence of IBE is not successful.
But I do not wish to end on such a negative note. In the next section, I will give a more positive account of Lipton’s results; because although I do not believe that they establish the validity of IBE, I do think that they allow us to see that explanation plays a more important (if non-epistemic) role in scientific method than the famous methodological systems – falsificationism, hypothetico-deductivism, Bayesianism – have allowed us to see.